Question

How many elements does each of these sets have where a and b are distinct elements?

(a) \({\bf{P}}\left( {\left\{ {{\bf{a,b,}}\left\{ {{\bf{a,b}}} \right\}} \right\}} \right)\)

(b) \(P\left( {\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}} \right)\)

(c) \(P\left( {P\left( \phi \right)} \right)\)

Short answer

(a) \(\left| {P\left( {\left\{ {a,b,\left\{ {a,b} \right\}} \right\}} \right)} \right| = 8\)

(b) \(\left| {P\left( {\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}} \right)} \right| = 16\)

(c) \(\left| {P\left( {P\left( \phi \right)} \right)} \right| = 2\)

Step-by-step solution

Power of a set

Given a set S, the power set of S is the set of all subsets of the set S.

Therefore, the power set of S is denoted by P(S).

To determine the number of elements of given set \(\left\{ {{\bf{a,b,}}\left\{ {{\bf{a,b}}} \right\}} \right\}\) (a)

The set \(\left\{ {a,b,\left\{ {a,b} \right\}} \right\}\)contain 3 elements \(a\) ,\(b\) and \(\left\{ {a,b} \right\}\).

The power set of \(\left\{ {a,b,\left\{ {a,b} \right\}} \right\}\) contains all possible subsets of \(\left\{ {a,b,\left\{ {a,b} \right\}} \right\}\).

The number of possible subsets is therefore equal to the product of the number of options for each element.

Therefore, the number of elements the set \(\left\{ {a,b,\left\{ {a,b} \right\}} \right\}\)has are

\(\begin{aligned}{c}\left| {P\left( {\left\{ {a,b,\left\{ {a,b} \right\}} \right\}} \right)} \right| = 2 \times 2 \times 2\\ = {2^3}\\ = 8\end{aligned}\)

Hence the number is 8.

To determine the number of elements of given set \(\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}\) (b)

The set \(\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}\)contain 4 elements \(\phi \) ,\(a\) ,\(\left\{ a \right\}\) and \(\left\{ {\left\{ a \right\}} \right\}\).

The power set of \(\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}\) contains all possible subsets of \(\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}\).

The number of possible subsets is therefore equal to the product of the number of options for each element.

Therefore, the number of elements the set \(\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}\)has are

\(\begin{aligned}{c}\left| {P\left( {\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}} \right)} \right| = 2 \times 2 \times 2 \times 2\\ = {2^4}\\ = 16\end{aligned}\)

Hence, the number is 16.

To determine the number of elements of given set \(P\left( {P\left( \phi  \right)} \right)\) (c)

The power set \(P\left\{ \phi \right\}\)contains all possible sets of \(\phi \).

And the only subset of \(\phi \) is the \(\phi \) itself.

The set \(\left\{ \phi \right\}\) contains one element.

There are two possible subsets of \(\left\{ \phi \right\}\): one is \(\phi \) and other is \(\left\{ \phi \right\}\).

Therefore, the number of elements the set \(P\left\{ \phi \right\}\) is 2.